‘Representing Interval Orders by Arbitrary Real Intervals’
This solves a representation problem
once left open by Fishburn and deals with some related themes.
Interval orders are a way to model
non-transitive indifference in comparison judgments
as well as temporal relations between events.
While so far only characterizations of their representability
by complete precedence vs. intersecting of
closed or open
(and bounded) real intervals have been known,
this paper presents
necessary and sufficient conditions for their representability
by arbitrary real intervals
as well as a new characterization for open representability.
Furthermore we, like Fishburn, consider natural restrictions
on representations and extend respective results of his.
Accordingly, we also deal with semiorders in the sense of Luce.
Interrelations of countability
(separability, representability) conditions
(“directly” in terms of interval orders)
reveal two redundancies in Fishburn’s representability conditions
and indicate more direct ways to his results.
The key to our results is a pair of generalizations
of Fishburn’s notion of “singularity”.
DVI (recommended): uncompressed, 198 KB | zipped, 82 KB
PDF: uncompressed, 411 KB | zipped, 325 KB
Keywords: interval order, semiorder, linear order, representation;
preferences/comparisons, economics/mathematical psychology/measurement.
Last update Mar 19, 2009